# Simple precedence parser

In computer science, a simple precedence parser is a type of bottom-up parser for context-free grammars that can be used only by simple precedence grammars.

The implementation of the parser is quite similar to the generic bottom-up parser. A stack is used to store a viable prefix of a sentential form from a rightmost derivation. Symbols $\lessdot$ , ${\dot {=}}$ and $\gtrdot$ are used to identify the pivot, and to know when to Shift or when to Reduce.

## Implementation

• Compute the Wirth–Weber precedence relationship table.
• Start with a stack with only the starting marker $. • Start with the string being parsed (Input) ended with an ending marker$.
• While not (Stack equals to $S and Input equals to$) (S = Initial symbol of the grammar)
• Search in the table the relationship between Top(stack) and NextToken(Input)
• if the relationship is ${\dot {=}}$ or $\lessdot$ • Shift:
• Push(Stack, relationship)
• Push(Stack, NextToken(Input))
• RemoveNextToken(Input)
• if the relationship is $\gtrdot$ • Reduce:
• SearchProductionToReduce(Stack)
• RemovePivot(Stack)
• Search in the table the relationship between the Non terminal from the production and first symbol in the stack (Starting from top)
• Push(Stack, relationship)
• Push(Stack, Non terminal)

SearchProductionToReduce (Stack)

• search the Pivot in the stack the nearest $\lessdot$ from the top
• search in the productions of the grammar which one have the same right side than the Pivot

## Example

Given the language:

E  --> E + T' | T'
T' --> T
T  --> T * F  | F
F  --> ( E' ) | num
E' --> E


num is a terminal, and the lexer parse any integer as num.

and the Parsing table:

EE'TT'F+*()num$E ${\dot {=}}$ $\gtrdot$ $\gtrdot$ E' ${\dot {=}}$ T $\gtrdot$ ${\dot {=}}$ $\gtrdot$ $\gtrdot$ T' $\gtrdot$ $\gtrdot$ $\gtrdot$ F $\gtrdot$ $\gtrdot$ $\gtrdot$ $\gtrdot$ + $\lessdot$ ${\dot {=}}$ $\lessdot$ $\lessdot$ $\lessdot$ * ${\dot {=}}$ $\lessdot$ $\lessdot$ ( $\lessdot$ ${\dot {=}}$ $\lessdot$ $\lessdot$ $\lessdot$ $\lessdot$ $\lessdot$ ) $\gtrdot$ $\gtrdot$ $\gtrdot$ $\gtrdot$ num $\gtrdot$ $\gtrdot$ $\gtrdot$ $\gtrdot$ $ $\lessdot$ $\lessdot$ $\lessdot$ $\lessdot$ $\lessdot$ $\lessdot$ STACK                   PRECEDENCE    INPUT            ACTION

$< 2 * ( 1 + 3 )$   SHIFT
$< 2 > * ( 1 + 3 )$     REDUCE (F -> num)
$< F > * ( 1 + 3 )$     REDUCE (T -> F)
$< T = * ( 1 + 3 )$     SHIFT
$< T = * < ( 1 + 3 )$       SHIFT
$< T = * < ( < 1 + 3 )$         SHIFT
$< T = * < ( < 1 > + 3 )$           REDUCE 4 times (F -> num) (T -> F) (T' -> T) (E ->T ')
$< T = * < ( < E = + 3 )$           SHIFT
$< T = * < ( < E = + < 3 )$             SHIFT
$< T = * < ( < E = + < 3 > )$               REDUCE 3 times (F -> num) (T -> F) (T' -> T)
$< T = * < ( < E = + = T > )$               REDUCE 2 times (E -> E + T) (E' -> E)
$< T = * < ( < E' = )$               SHIFT
$< T = * < ( = E' = ) >$                REDUCE (F -> ( E' ))
$< T = * = F >$                REDUCE (T -> T * F)
$< T >$                REDUCE 2 times (T' -> T) (E -> T')
$< E >$                ACCEPT